The continuing goal of reducing critical dimensions (CD) in semiconductor manufacturing is putting increasing pressure on optical lithography. For further information, See "Methods and Apparatus for Integrating Optical and Interferometric Lithography to Produce Complex Patterns", Ser. No. 09/273,399, filed on Mar. 22, 1999 with inventors Brueck, et al and U.S. Provisional Application No. 60/111,340 filed on Dec. 7, 1998 entitled "Arbitrary Lithographic Patterns" with inventors S. R. J. Brueck, Xiaolan Chen, Andrew Frauenglass and Saleem H. Zaidi. The entire contents of the foregoing are incorporated herein by reference.
The diffraction-limited resolution of optical lithography is usually expressed in a simplified form by the Rayleigh resolution criteria, viz. ##EQU1##
Where .lambda. is the optical wavelength of the exposure, NA is the lens numerical aperture and .kappa..sub.1 is related to process latitude. Throughout the development of optical lithography for semiconductor manufacturing, NAs have increased to about 0.65 and wavelengths have decreased to about 248 nm, and significant effort is being placed on a further reduction of wavelengths to 193 nm and possibly 157 nm. However, both of these trends are facing fundamental limits. For example, aberrations increase in proportion to a high power of the NA, thereby making further increases problematic. Moreover, available optical materials typically limit the wavelength for a transmissive optical system. Thus, attention has turned to reductions in .kappa..sub.1.
The Rayleigh resolution criteria of Equation 1 demonstrates the frequency space constraints of an imaging optical system. Particularly, a lens is a low pass filter [J. Goodman, Introduction to Fourier Optics, 2nd edition, (McGraw Hill, New York, 1996)] with a bandwidth of .function..sub.opt =NA/.lambda. which results in Eq. 1. The traditional Rayleigh criterion of an intensity dip of only about 20% between two resolvable peaks results in a minimum .kappa..sub.1 of 0.61. Using partially coherent illumination and high contrast photoresists, a satisfactory process latitude can be achieved at a .kappa..sub.1 of 0.61 which gives a minimum CD.about..lambda. for modern, high-NA lenses. However, it is not possible to decrease .sub.1 further in conventional imaging without losing the image entirely.
The limitations on decreasing .kappa..sub.1, along with the limitations on decreasing the other factors of Equation 1, have led to the exploration of a number of resolution-enhancement techniques (RET). Moreover, the high cost of enabling post-optical lithography technologies such as: x-ray; e-beam; ion-beam and extreme-ultraviolet lithography provides an incentive to extend optical lithography as far as possible. The RETs currently include, for example, optical proximity correction (OPC), off-axis illumination (OAI), phase-shift masks (PSM) and imaging-interferometric lithography (IIL). Each of these RETs improves the image quality and allows printing of lithographic features at lower .kappa..sub.1. While much of the discussion and analysis of these techniques has centered on the real-space (image) implications, there is, of course, a corresponding view that emphasizes the spatial frequency content of the image. As discussed below, one embodiment of the present invention preferably takes advantage of the spatial frequency effects of some of these techniques.
For a recent review of RET, see, for example, M. D. Levenson. "Wavefront engineering from 500 mn to 100 nm CD." Proc. SPIE 3048. 2-13 (1997). L. W. Liebmann, B. Grenon, M. Lavin, S. Schomody, and T. Zell, "Optical proximity correction a first look at manufacturability," Proc. SPIE 2322, 229-238 (1994)). For further information on OPC, see, for example, P. D. Robertson, F. W. Wise, A. N. Nasr, and A. R. Neureuther, "Proximity effects and influence of nonuniform illumination in projection lithography," Proc. SPIE 334, 14 (1982), L. W. Liebmann, B. Grenon, M. Lavin, S. Schomody. and T. Zell, "Optical proximity correction, a first look at manufacturing," SPIE 2322. 14th Annual BACUS Symposium on Photo-mask Technology and Management, 229-238(1994); J. F. Chen, T. Laidig, K. E. Wampler, R. Caidwell, "Full-chip optical proximity correction with depth of focus enhancement", Microlithography World, Summer, 5-13(1997). For further information on OAI, see, for example, M. Noguchi, M. Muraki, Y. Iwasaki, and N. Magome. "Sub-half micron lithography system with phase-shifting effect," Proc. SPIE 1674, 92-104 (1992). K. Tournai, H. Tanabe, H. Nozue, and K. Kasama. "Resolution improvement with annular illumination," Proc. SPIE 1674, 753-764 (1992); K. Kamon, T. Miyamoto, Y. Myoi, H. Nagata, M. Tanaka and K. Hone, "Photolithography system using annular illumination," Jpn. J. Appi. Phys., Vol. 30, No. 1 1B. 3021-3029(1991); N. Shiraishi, S Hirukawa, Y. Takeuchi and N. Magome, "New imaging technique for 64 M-DRAM," SPIE 1674, Optical/Laser Microlithography V, 741-752(1992). For further information on PSM, see, for example, M.D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase shifting mask," IEEE Trans. Electron Devices. ED-29, 1828-1836(1982), M. D. Levenson, D. G Goodman, S. Lindsey, P. Bayer and H. Santini, "The phase shifting mask II: imaging simulations and submicron resist exposures," IEEE Trans. Electron Devices ED-31, 753-763 (1984), H. Watanabe, H. Takenaka, Y. Todokoro, and M. lnoue, "Sub-quarter-micron gate pattern fabrication using a transparent phase shifting mask," J. Vac. Sci. Technol. B9, 3172 (1991)), H. Jinbo and Y. Yamashita, "Improvement of phase-shifter edge line mask method," Jpn. J. Appl. Phys. Vol. 30, 2998-3003(1991). For further information on IIL, see, for example, X. Chen and S. R. J. Brueck, "Inaging interferometric lithography for arbitrary patterns," Proc. SPIE 3331, 214-22y (1998). X. Chen and S. R. J. Brueck, "Imaging interferometric lithography--a wavelength division multiplex approach to extending optical lithography," J. Vac. Sci Technol. The entire contents of all of the foregoing references are incorporated herein by reference.
More particularly, the following sets forth a spatial frequency analysis of optical lithography RETs. With respect to Fourier optics, the basic goal is typically to increase the high-frequency content of the image. For a free-space optical transmission medium, the highest attainable spatial frequency is .function..sub.IL =2/.lambda., corresponding to counterpropagating plane waves at grazing incidence to the wafer. For dense features, the resulting CD is .lambda./4. This limit is relatively easy to approach experimentally by interferometric lithography using a small number of plane waves resulting in a repetitive pattern such as a 1D grating or a 2D dot array. See, for example, X. Chen, Z. Zhang, S. R. J. Brueck, R. A. Carpio and J. S. Petersen, "Process Development for 180-nm Structures using Interferometric Lithography and I-Line Photoresist," Proc. SPIE 3048, 309-318 (1997), which is incorporated herein by reference. Useful patterns contain a very large number of spatial frequencies, wherein corners and edges often include frequencies beyond this limiting value even for dense patterns. While imaging systems record many spatial frequencies at once, imaging systems often only record the spatial frequencies within the .about..function..sub.opt bandwidth of the optical system. However, RETs allow extension of this bandwidth towards 2.function..sub.opt (OPC, OAI, PSM) and to the limiting value of .function..sub.IL (IIL).
For coherent illumination (see, for example, J. Goodman "Introduction to Fourier Optics" 2nd edition, (McGraw-Hill, New York, 1996)), the optical system passes all of the (electric field) frequency components of the diffraction with a unity coherent modulation transfer function (MTF) up to a frequency of .function..sub.opt, e.g. EQU T.sub.coh (.function.)=1 for .function.&lt;.function..sub.opt EQU =0 for .function.&gt;.function..sub.opt (2)
The intensity aerial image contains frequencies up to 2.function..sub.opt as a result of the square-law photoresist response, but these frequencies do not necessarily correspond to findamental frequencies present in the original image. For incoherent imaging, the modulation transfer function (see, for example, J. Goodman "Introduction to Fourier Optics" 2nd edition, (McGraw-Hill, New York, 1996)) for the frequency space amplitudes is: ##EQU2##
which extends out to 2.function..sub.opt, but it decreases rapidly with .function. and is 0.5 at 0.8.times..function..sub.opt and only 0.04 at 1.5.times..function..sub.opt. For partially coherent illumination, which is predominantly used for manufacturing, a simple, pattern-independent expression for the MTE is not available, but the essence of these limits is retained, thus, conventional imaging provides a useable MTF only out to frequencies .about..function..sub.opt.
Considering the foregoing explanation, a qualitative analysis of the spatial frequency basis of each of the RETs will now be set forth. For OPC (partially coherent or incoherent illumination), sub-resolution features (corresponding to high spatial frequencies) are added to the mask to boost the magnitudes of the high spatial frequencies in the Fourier transform beyond those of the image. However, adding the high spatial frequencies to the mask significantly increases the mask's complexity. The magnitudes of the high frequencies are attenuated by the optical system but are still larger than they would have been without the OPC, resulting in an image closer to the desired pattern, but no longer simply related to the mask pattern. It is intuitively clear that this approach can deal with frequencies near .function..sub.opt where the T.sub.inc is .about.0.5 but is inadequate to add substantial frequency content out towards 2.function..sub.opt where T.sub.inc is very low. Because of these restrictions in frequency space. OPC extends .kappa..sub.1 to 0.5-0.4 but not below.
In OAI, the (coherent or partially coherent) illumination beam is tilted away from the optical system normal to allow higher spatial frequencies to pass through the optical system. These tilts are restricted to a fraction .eta.&lt;1 of .function..sub.opt in order to transmit the zero-order beam through the optical system and the maximum spatial frequency is given by (1+.eta.).function..sub.opt &lt;2.function..sub.opt. As will be shown it more detail below, there are typically significant spectral overlaps that further emphasize the lower spatial frequencies, restricting .kappa..sub.1 to .about.0.43. If these overlaps are eliminated, OAI can be extended to a lower .kappa..sub.1.
For PSM, the mask is often drastically modified (to a 3D object) to enhance the amplitudes of the fundamental frequency components (out to .function..sub.opt) while reducing the amplitude of the dc component of the electric-field. Because of the square-law resist response, this dramatically boosts the quadratic image intensity components between .function..sub.opt and 2.function..sub.opt allowing .kappa..sub.1 's of 0.35. As was the case for OPC, the simple relationship between the mask and image pattern is lost requiring extensive design and mask fabrication efforts.
IIL is a newly introduced imaging concept that is closely related to OAI, with the major difference that the offsets are chosen beyond .function..sub.opt, requiring a separate optical system to arrange for the zero-order reference beam at the wafer. For continuous coverage along the principal (.function..sub.x,.function..sub.y) frequency axes, the maximum frequency at the wafer is as high as 3.function..sub.opt, allowing .kappa..sub.1 's of 0.2. Additional exposures allow further extensions to the fundamental optics linear systems limit of 2/.lambda.. As with OAI, the mask and image patterns are identical (except for magnification scaling) and traditional binary chrome-on-glass masks are used.
With respect to modeling, a simple, diffraction-limited Fourier optics model serves as the basis for understanding the frequency space impacts of each of the RET techniques. The starting point is the Fourier transform of the mask function. For simplicity, a 1.times. magnification is assumed; this does not affect any of the conclusions, but simplifies the equations. For coherent illumination the electric field (within a scalar electromagnetic model) at the wafer is given by: ##EQU3##
where M(.function..sub.x,.function..sub.y) is the Fourier transform of the mask pattern, the dc-term has been included explicitly outside of the summation and the usual normalization, M(0,0)=1, has been applied. Because the zero frequency term has been pulled out, the summation does not include the (0,0) term. This normalization is appropriate for all cases except for a phase shift mask with equal clear 0.degree. and 180.degree. areas [e.g.: M(0,0)=0]. The photoresist responds to the intensity given by: ##EQU4##
where the dc term is again excluded from the summations. Remembering that M(.function..sub.x,.function..sub.y) is the Fourier transform of the mask transmission function, the first two complex-conjugate terms in Eq. (5) are the Fourier transform of the mask filtered by the optical system MTF and inverse transformed back to real space. These terms give an accurate representation of the mask pattern up to the lens cutoff frequency, .function..sub.opt. The constant term is a uniform exposure since the intensity cannot be negative. The final term, quadratic in the Fourier amplitudes, is the intensity autocorrelation of the mask image with frequencies extending to 2.function..sub.opt. None of the frequency component amplitudes of this autocorrelation correspond directly (i e. linearly) to the desired image. The impact of the frequency components can be beneficial and improve the contrast. For example, for a grating structure at a pitch .function..sub.opt /2&lt;.function..sub.g &lt;.function..sub.opt, the square term at 2.function..sub.g adds a second-harmonic intensity component that was cutoff by the lens, but not necessarily with the same coefficient as the image transform. These terms can also be detrimental, for example, occurring at frequencies not present in the original mask transform. For compactness in what follows, the first two terms, linear in the Fourier amplitudes, will be referred to as `linear` while the autocorrelation terms will be noted as `quadratic.` As will be discussed in more detail below, the quadratic terms are the basis of PSM enhancements.
The mathematics is somewhat simpler for incoherent illumination (see, for example, J. Goodman "Introduction to Fourier Optics" 2nd edition, (McGraw-Hill, New York, 1996)), viz. ##EQU5##
The image is simply the transform of the mask multiplied by the incoherent MTF and retransformed back to real space.
The final step in evaluating the image is to evaluate the response of the photoresist to the calculated aerial image. For all of the calculations presented herein, a simple threshold function is used. The resist remains unexposed for all intensities below a threshold level and is fully exposed for intensities above this level. This is a vastly oversimplified model of actual resist behavior. However, it allows rapid calculation and keeps the focus on the optics effects. Notwithstanding this simplification, excellent qualitative agreement is obtained between the model calculations presented here and recent experiments. See, for example, X. Chen and S. R. J. Brueck, "Imaging interferometric lithography for arbitrary patterns," Proc. SPIE 3331, 214-22y (1998). X. Chen and S. R. J. Brueck, "Imaging interferometric lithography--a wavelength division multiplex approach to extending optical lithography," J. Vac. Sci Technol. (to be published).
With respect to the ideal linear systems limits of optics, the formalism of Eqs. 4 and 5 provides a simple way to evaluate this limit. As noted above, the linear terms contain the pattern information. FIG. 1 shows a simple calculation of the limiting optics capability including all of the linear terms up to the .function..sub.IL free-space optics bandwidth limit and ignoring the quadratic contributions for a test pattern including a series of five nested "ells" with both dense and isolated features as well as a large feature included to ensure a wide range of spatial frequencies. All of the calculations presented are for a 193-nm exposure wavelength. Straightforward wavelength scaling increases these CDs by a factor of 1.9 for i-line and decreases them by 0.81 for a 157-nm exposure wavelength. The figure shows an area of 24.times.24 CD; the actual pattern, which is identical to that used in the experiments reported elsewhere (see, for example, X. Chen and S. R. J. Brueck, "Imaging interferometric lithography for arbitrary patterns," Proc. SPIE 3331, 214-22y (1998)), repeats on a larger grid of 40.times.40 CD.sup.2 to ensure an adequate sampling of frequency space. The modeled image is a good realization of the desired pattern for a CD of 69 nm (/2.8) with only some rounding of the nominally square line ends showing the impact of the frequency limits of the transmission medium. At a CD of 56 nm (/3.5), some linewidth variation and a dense-isolated line bias is evident. At a CD of 48 nm (/4), the pattern has developed the strong linewidth fluctuations that are characteristic of a coherent illumination exposure just at the frequency cut-off where the fundamental frequency (1/2CD) of the line-space pattern is transmitted, but the higher frequencies necessary to define the edges and comers of the pattern are eliminated by the bandwidth constraint. The definition of the corners of the `ells` is also impacted and a hole has appeared in the upper left corner of the large box. The required high-spatial frequency coverage is roughly given by ellipses centered around .+-.1/2CD with a halfwidth of (1/2nCD) where n is the number of nested lines (5 in this case) along the frequency axis and a halfwidth of 1+L /L where L is the length of the line in the direction perpendicular to the frequency axis. Based on this qualitative calculation, the minimum CD for the present pattern at a 193-nm exposure wavelength is 58 nm .about./3.3 in good agreement with the detailed modeling. The small-scale jagged edges on the figures are an artifact of the pixelization of the calculation (CD/12). It is stressed that this calculation is only intended to set a standard with which to compare the capabilities of the optical enhancement techniques. As such, there is no physical optical system associated with the calculation and no way to eliminate the quadratic terms in a real experiment. Nevertheless, as shown below, the model calculations approach this idealized limit quite closely showing that there is a potential to extend optical lithography well beyond its current implementations.
With respect to optical proximity correction (OPC), the basic idea is to add sub-resolution features to boost the amplitudes of the high frequency image components so that after the attenuation of the optical system, the high frequency components are closer to required levels. FIG. 2 shows a simple example for 193-nm incoherent illumination with a 0.65 NA optical system and a 132-nm CD (.kappa..sub.1 =0.44). The top left panel shows the non-OPC mask pattern. It is modeled as repetitive in both the x- and y-directions with a period of the outline of the large box. This makes the summations discrete and limited to a small number of frequencies enabling rapid image calculations. The bottom left shows the printed image using the uncorrected mask. There is significant rounding of all of the corners and foreshortening of the isolated lines. The top right panel shows a mask pattern with an empirically derived OPC including line extensions (nailheads) for outside corners and missing areas (mouse bites) for inside corners. Note that the OPC adds significantly to the mask complexity. The impacts of these changes extend for a scale of .about..lambda. on the image, making the corrections nonlocal and requiring considerable design and computation to settle on an optimum correction. Finally, the bottom right panel shows a modeled image with OPC that is much closer to the desired pattern.
The impacts of OPC in frequency space are shown in FIG. 3. The top panels show the positive (left) and negative (right) differences between the amplitude Fourier transforms of the OPC and the uncorrected masks just at the mask plane. The large circle represents the .function..sub.IL limit of optics. The two dotted circles are at radii of .function..sub.opt (corresponding to the maximum frequency covered in the linear terms) and 2.function..sub.opt (indicating the maximum frequency available from the quadratic terms). This convention will be retained for all subsequent frequency-space illustrations. Note that the sub-resolution OPC adds spectral intensity all the way out to .function..sub.IL. The bottom panels show similar plots for the amplitude Fourier transforms of the aerial image at the wafer plane. The optical system MTF eliminates the high frequency components and limits the changes in the final image to just beyond .function..sub.opt. This frequency restriction limits the possible improvement over conventional photolithography.
As such, OPC is applicable for incoherent or partially coherent illumination. The coherent illumination MTF already passes all of the frequency components up to .function..sub.opt, with a unity transfer function and eliminates all of the components above .function..sub.opt. Adding frequency components above .function..sub.opt with OPC has no impact on the final image for coherent illumination.
With respect to off-axis illumination (OAI), numerous variants on off-axis illumination have been investigated (For a recent review, see M. D. Levenson. "Wavefront engineering from 500 nm to 100 nm CD." Proc. SPIE 3048. 2-13 (1997)). In general, for Manhattan geometry patterns, quadrupole illumination is preferred. The three parameters that define the geometry for hard quadrupole apertures are the offset angle, the orientation of the quadrupole relative to the pattern x-axis and the diameter of the aperture (the partial coherence). For small aperture diameters relative to the offset angle, the quadrupole illumination is equivalent to an incoherent summation over four independent off-axis, coherent illumination exposures. This is the simplest configuration that retains the essential physics of the off-axis illumination and is most closely related to IIL. This analysis will be restricted to this simple, limiting case.
Equations 4 and 5 must be modified to account for the off-axis illumination. For each exposure, they become: ##EQU6##
where .function..sub.off =.eta..function..sub.opt is the offset spatial frequency which is necessarily less than .function..sub.opt and .sigma. is the orientation of the particular hole of the quadrupole relative to the pattern x-axis. The first term in Eq. 7 is the zero-order diffraction (transmission) shifted because of the off-axis illumination. The same shift occurs in all of the other diffracted orders as well. Since the offset appears in the argument of the MTF the region of frequency space covered is shifted from the origin. As the offset is the same for all spatial frequencies, it is eliminated from the exponential terms on squaring to the intensity and the only impact is through the MTF functions. Because a real image must have a complex conjugate relationship between frequency terms reflected through the origin (cf. Eq. 8) each exposure covers two circles in spatial frequency space of radius .function..sub.opt and centered along the offset direction at .+-..function..sub.off. These circles are overlapped in the vicinity of the origin out to frequencies of .+-.(1-.eta.).function..sub.opt along the offset direction. The opposing hole (.sigma.{character pullout}.sigma.+.pi.) of the quadrupole gives the identical result for the intensity and serves primarily to preserve the telecentricity of the exposure without adding any additional image information at focus. The remaining two holes give patterns with the offset along an orthogonal axis.
FIG. 4 shows the final frequency space coverage. There are spatial frequency regions, clustered at low frequency, where the same spectral intensities are covered in multiple exposures. This overlap leads to a frequency dependent MTF and results in a roll-off in the MTF towards higher frequencies. The density of the fill in each region of FIG. 4 is proportional to the MTF in that region. The highest MTF occurs at the frequency space origin where all four exposures contribute to the image frequency components. The axes labeled .function..sub.x (.function..sub.y) correspond to a /4 rotation of the quadrupole relative to the pattern x (y) axes. Along this direction the overlap decreases from four circles to two circles (MTF{character pullout}0.5) at a frequency .function..sub.1 and drops to no coverage (MTF{character pullout}0) at a frequency .function..sub.2. Similarly, along the axes labeled .function..sub.x' (.function..sub.y') corresponding to alignment of the quadrupole axes with the pattern axes, the coverage drops to three circles .function..sub.1' (MTF{character pullout}0.75), to one circle at .function..sub.2' (MTF{character pullout}0.25) and to no coverage (MTF{character pullout}0) at .function..sub.3'. In terms of and .function..sub.opt these frequencies are given by: ##EQU7##
The contribution to the MTF from only the first two (`linear`) terms in Eq. 8 along both the .function..sub.x and .function..sub.x' axes is plotted in FIG. 5 along with the MTF of an incoherent exposure for reference. It is worth noting that the decrease in the incoherent MTF arises from a very similar effect of multiple counting in frequency space. Incoherent illumination can be evaluated as a summation over many independent coherent illumination exposures covering all possible incident azimuthal and polar angles, each exposure covers a pair of circles in frequency space, with appropriate cutoffs for frequencies beyond the lens numerical aperture. The derivation of the incoherent MTF, Eq. 3, involves integrating over these multiple exposures in much the same way that result of FIG. 5 was obtained. The results of FIG. 5 are in excellent agreement with a much more detailed calculation, including effects of partial coherence and defocus, by Tounai et al. (K Tournai, H. Tanabe, H. Nozue, and K. Kasama "resolution improvement with annular illumination," Proc. SPIE 1674, 753-764 (1992)).
The quadratic terms also impact the final image. Thus, it is not possible to derive an image-independent, intensity MTF including both the linear and quadratic terms for the case of coherent illumination. Using spatial filters, there are many non- unique ways to "tile" frequency space, i.e. to eliminate these overlaps and assure that the linear terms are counted equivalently out to the limits of the pattern. However, these tilings have different quadratic terms and this can significantly impact the final printed pattern as will be discussed in more detail below.
Inspection of FIGS. 4 and 5 immediately illuminates the tradeoffs involved in rotation of the quadrupole relative to the pattern axes. For a 45.degree. rotation (.function..sub.x,.function..sub.y), the MTF has dropped less dramatically at moderate spatial frequencies (0.5 at .function..sub.opt) compared with a 0.degree. rotation (.function..sub.x',.function..sub.y') where the MTF is only 0.25 at .function..sub.opt, but it does not extend as far in frequency space. As.fwdarw.1, .function..sub.2.fwdarw./2.function..sub.opt while .function..sub.3'.fwdarw.2.function..sub.opt. These differences have important impacts on the lithographic capability as the CD is decreased, requiring higher spatial frequencies to define the image.
FIG. 6 shows the real-space modeling results for printing the same test pattern that was used to evaluate the ideal optics limits at a CD of 130 nm using a 193-nm exposure and an NA=0.65 optical system (.sub.1 =0.44). The top panel is the result of a conventional, normal incidence illumination, coherent exposure. The high spatial frequencies that define the dense line-space pairs are not transmitted by the optical system and the resulting image has merged into a single amorphous feature. The bottom panel shows the result of a quadrupole OAI exposure =0.75 and =45.degree.. The line-space pairs are now well resolved and, except for some dense/isolated bias (the horizontal and vertical isolated lines are narrower than the dense lines) and rounding of the sharp comers, the pattern is captured faithfully.
FIG. 7 shows frequency space views of these exposures. It is easier to interpret spectral differences rather than to compare the full spectra. The top panel shows the spectral intensities added by OAI. As expected, the added features are concentrated around 1/(2CD) in both the .function..sub.x and .function..sub.y directions. The middle panel of FIG. 7 shows the spectral intensities suppressed by OAI compared with conventional OL. These are all &lt;.function..sub.opt. The distribution of spatial frequency content along the .function..sub.x and .function..sub.y axes is evident. Finally, the bottom panel shows the difference between the ideal linear systems limit of optics and the OAI exposure. Significant additional spectral intensity is needed around the fundamental frequency [1/(2CD)] of the line-space pattern. The impact of this decrease in intensity is both in the details of the pattern as can be seen in FIG. 6 and in the process latitude. The calculated aerial images clearly show substantial differences in process latitude (i.e. in the depth of the intensity modulation between lines and spaces) for the various exposure strategies. This is the limit to the dense CD that can be printed in this optical configuration. For smaller CDs important frequency components are lost as the effective MTF.fwdarw.0 as shown in FIGS. 4 and 5. It is useful to make this limit somewhat quantitative. The maximum spatial frequency covered along the principal axes is .function..sub.2 (0.75)=1.38.function..sub.opt. As discussed above, the maximum spatial frequency required is .about.1.2.times.[1/(2CD)], giving a minimum CD of .about.129 nm for a 193-nm exposure in this off-axis configuration.
FIGS. 8 and 9 show comparable real-space and frequency-space views for the same pattern and optical system with a smaller CD of 105 nm, =0.75 and .sigma.=0.degree. (.sub.1 of 0.35). The smaller CD could be resolved because of the greater frequency coverage of the 0.degree. geometry. However, the dense-isolated line bias is much more pronounced with the longer horizontal isolated line even being broken; there is also some distortion of the large box. The reasons for these effects are evident in the frequency space plots. The frequencies added by OAI are further out in frequency space, corresponding to the smaller CD, but the intensities added (number of contours) are weaker because of the lower effective MTF. Similarly, the bottom panel that shows the spectral additions necessary to get to the limit of optics are much more pronounced than for the 45.degree. case. Here again, a very simple calculation serves to evaluate the minimum dense CD that can be printed in this geometry. The calculation is identical to that above with f3'( ).fwdarw.1.75 replacing .function..sub.2 giving .about.102 nm at a 193-nm exposure wavelength.
With respect to phase-shift masks (PSM), many variations of phase-shift masks have been introduced (For a recent review, see M. D. Levenson. "Wavefront engineering from 500 nm to 100 nm CD." Proc. SPIE 3048. 2-13 (1997)) including strong (alternating aperture) and weak (attenuated phase-shift). For the purposes of this discussion, a simple chromeless phase-shift mask will be used to illustrate the frequency space impacts of phase shift masks. For many patterns, a chromeless phase-shift mask introduces extra pattern lines which require a second elimination exposure. The basic idea of the phase-shift mask is understood simply by reference to Eqs. 4 and 5. The phase shift enhances the relative magnitudes of the non-dc Fourier coefficients and, hence, the linear terms, but the increase in the quadratic terms is much greater emphasizing these high frequency contributions to the final image.
FIG. 13 shows the simple mask set used for the modeling. The goal is to print a "u" shape with arms at the CD and separated by a CD. The two masks consist of a phase-shift mask with a phase shift in a rectangular region 2.5 CD high that prints the outline of the rectangle, and a conventional, binary chrome-on-glass mask to eliminate the second half of the rectangle leaving the "u". Again, these are taken as repetitive patterns with repeat lengths given by the boxes containing the masks. FIG. 14 shows the electric field strength immediately following the phase-shift mask along the cut indicated by the dotted line in FIG. 13. Two curves are shown. The dash-dot curve (PSM) corresponds to the phase-shift mask. The solid curve (Cr) is the field profile for a conventional chrome-on-glass mask of the same shape for reference. Since both curves are simple square waves, the Fourier transforms are both sinc functions, sampled at the .function..sub.x - and .function..sub.y -frequencies defined by the periodicity. Two impacts of the phase shift are immediately evident. The field variation for the phase shift mask (1 to -1) is twice as large as that for the conventional mask (1 to 0) doubling the Fourier magnitudes, and the dc Fourier coefficient, corresponding to the integral over the field, is smaller for the phase shift mask because of the negative field in the phase shift region. Combined, these two effects strongly increase the importance of the quadratic terms in the imaging (Eq. 8). For this example, one skilled in the art will appreciate that all of the non-zero frequency terms are increased by 2.2 (normalized to M.sub.00 =1), and, of course, the quadratic coefficients are increased by 2.2.sup.2 =48.
FIG. 15 shows the printed pattern modeled for a CD of 106 nm using a coherent exposure with a NA=0.65 lens at a 193-nm exposure wavelength (.kappa..sub.1 =0.36). The extra high-frequency content results in printing the outline of the rectangle. The right-hand side of the rectangle was eliminated with a second blocking exposure. The rounding of the pattern corners results since the linear frequencies that define the corner for a conventional exposure are eliminated by the optical system, hence the quadratic frequencies that give the outline are also eliminated. Based on this modeling, .kappa..sub.1 of .about.0.4-0.35 are possible using phase-shift techniques.
With respect to imaging interferometric lithography (IIL), IIL introduces a zero-order diffraction at the wafer plane. In contrast, the OAI offset angles are constrained to be within the lens NA. The reason for this constraint is clear from Eq. 8, if the offset were to be set outside of the NA, the zero-order diffraction would not be transmitted by the optical system [T.sub.coh (-.function..sub.off cos .sigma.,-.function..sub.off sin .sigma.).fwdarw.0] and the linear terms in. the intensity vanish leaving only the quadratic terms. As noted above, the principal image information is in these linear terms, so that OAI with a conventional mask is simply not possible at .function..sub.off &gt;.function..sub.opt. IIL resolves this issue by introducing a second beam path around the optical system to reintroduce the zero-order diffraction at the wafer plane. This allows much larger offset angles, and, thereby, much larger spatial frequencies, resulting in smaller CDs or, equivalently, smaller .kappa..sub.1' s. For a telecentric lens geometry, where the mask and wafer planes are necessarily normal to the optical axis, the maximum offset angle is grazing incidence (.function..sub.off.fwdarw.1/), giving a maximum spatial frequency of (1+NA/.lambda. for NA&gt;0.5 or .sup.3NA /.lambda. for NA .alpha. 0.5 compared with 2+L NA/.lambda. (at .eta.=1, .sigma.=45.degree.) or 2NA/.lambda. (at .eta.=1, .sigma.=0.degree.) for OAI. As in the case of OAI, telecentricity of the entire optical system (as opposed to just the imaging lens) can be recovered at the expense of additional optical complexity by introducing a second, incoherently-related, reference from the opposing side. Just as is the case for OAI, the multiple exposures can be simultaneous as long as the incident beams are incoherent, for example using multiple, independent laser sources. For tilted mask and wafer geometries, the ultimate linear systems frequency limit of optics .function..sub.IL =2/.lambda. can be reached. However, the extra difficulty of non-telecentric lenses and imaging systems probably makes this a less desirable alternative, at least for initial applications.
A time-domain communications analogy, wavelength division multiplexing (WDM), provides additional perspectives on the physics of IIL. WDM is used when the available bandwidth of a transmission medium (e.g. free-space or an optical fiber) is much larger than the electronic systems bandwidth used for modulating and demodulating the signal. Multiple frequency- (or wavelength-) offset carriers can then carry independent data streams. Modulation and demodulation for each data stream is accomplished by nonlinear mixing. The entire data stream can be reassembled after the multiple, independent transmissions. For IIL a very similar process occurs in space rather than in time. The ultimate bandwidth of optics is .function..sub.IL, the optical system limits the bandwidth for a single transmission to .function..sub.opt =NA.ident..left brkt-bot.&lt;&lt;.function..sub.II... The square-law conversions from intensity to electric field and back provide the nonlinear mixing. Off-axis illumination down-shifts the spatial frequencies within a specific area of frequency space into the optical system bandwidth and the interference with the reference beam resets the spatial frequencies after the optical system. Only a single-sideband is collected by the lens, the nonlinear mixing restores both sidebands around the (DC) image carrier frequency. The photoresist accumulates the total image information from the multiple exposures.
In summary, nonlinear processes can further extend the frequency space available to optics. As a trivial example, the aerial image for simple two-beam interference is 1+cos(4.pi. sin .theta./.lambda.). After the nonlinear development step, square patterns with nearly vertical sidewalls are formed with a Fourier decomposition of ##EQU8##
where .xi..lambda./(2 sin .theta.) is the linewidth and .lambda./(2 sin .theta.) is the period. This extends to much higher frequencies (n&gt;1) than does the aerial image. As demonstrated by the modeling, the linear systems limits of optics, which can be reached by the various RETs, extend well beyond today's practical limits. Optimization of the lithographic response and use of nonlinear processes will further extend the practical scales for optical techniques. There is substantial mileage remaining to allow further reductions in scale while still maintaining the parallel processing and manufacturing capability that have made optics the dominant lithographic manufacturing technique.